ENG - Derivation of all linear transformations - 2

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Derivation of All Linear Transformations that Meet the Results of

Michelson-Morley’s Experiment and Discussion of the Relativity

Basics

Roman Szostek

Rzeszow University of Technology, Department of Quantitative Methods, Rzeszow, Poland [email protected]

Abstract: The paper presents a formal proof that the mathematics on which the Special Theory

of Relativity (STR) is based is currently misinterpreted. The evidence is based on an analysis

of the importance of parameter e(v). Understanding the meaning of this parameter was

achieved by analyzing the general form of transformation, for which the Lorentz

transformation is only a special case. If e(v) ≠ 0 then the clocks in inertial systems are

desynchronized. Measurements, e.g. one-way speed, using such clocks do not give real values.

The article shows that there are infinitely many different transformations in which one-way

speed of light is always equal to c. The Lorentz transformation is only one of those infinitely many

transformations.

In this article, the whole class of linear transformations of time and coordinate was derived.

Transformations were derived on the assumption that conclusions from Michelson-Morley’s and

Kennedy-Thorndikea’s experiments are met for the observer from each inertial frame of reference,

i.e. that the mean velocity of light in the vacuum flowing along the way back and forth is constant. It was also assumed that there is at least one inertial frame of reference, in which the velocity of light in a vacuum in each direction has the same value c, and the space is isotropic for observers from this distinguished inertial frame of reference (universal frame of reference).

Derived transformations allow for building many different kinematics according to

Michelson-Morley’s and Kennedy-Thorndikea’s experiments.

The class of transformations derived in the study is a generalization of transformations

derived in the paper [10], which consists in enabling non-zero values of parameter e(v). The idea of

such a generalization derives from the person, who gave me this extended transformations class for

analysis and publication.

Keywords: coordinate and time transformation; kinematics; universal frame of reference; one-way

speed of light; anisotropy of cosmic microwave background

1. Introduction

The class of transformations derived in this article is a generalization of transformations

derived in the paper [10]. In that paper all linear transformations that are possible for the parameter

e(v) = 0 were derived. In the paper [11] one of these transformations was analyzed.

Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 5 August 2019 doi:10.20944/preprints201908.0048.v1

© 2019 by the author(s). Distributed under a Creative Commons CC BY license.

https://doi.org/10.20944/preprints201908.0048.v1

http://creativecommons.org/licenses/by/4.0/

Derivation of all linear transformations that meet the results of Michelson-Morley’s experiment and

discussion of the relativity basics - Szostek Roman

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It is a common belief in the contemporary physics that the Michelson-Morley [3] and

Kennedy-Thorndike [1] experiments proved that the velocity of light is absolutely constant and that

there is no universal frame of reference called the ether. Based on the analysis of these experiments,

the Lorentz transformation, on which the Special Theory of Relativity is based, was derived. It is

currently considered that the Special Theory of Relativity is the only theory of kinematics of bodies

which correctly explains the Michelson-Morley experiment and all other experiments in which the

velocity of light is measured.

It turns out that the velocity of light in one direction (momentary) has never been accurately

measured. In all accurate measurements of the velocity of light, only the average velocity of light

traveling the path along the closed trajectory was measured. In order to measure the velocity of

light, light had to return to the measuring device. In the simplest case, light was sent to a mirror and

back as was done in experiments by Armand Fizeau in 1849 and by Jean Foucault in 1850. The

same happens in Michelson-Morley and Kennedy-Thorndike experiments in which sources of light

after being reflected by mirrors return to the source point. From these experiments, it is clear that

the average velocity of light traveling the path to and back is constant, and not that the velocity of

light in one direction (momentary) is constant.

There are publications in which numerous coordinate and time transformations [2], [4], [5],

[6], [14] are presented. In this article all possible linear transformations (without turnover) are

derived. Derivation presented in this article is based on the postulate of average velocity of light,

and not on clock synchronization. From the presented analysis it results that there are infinitely

many coordinates and time transformations which are in accordance with the results of Michelson-

Morley’s experiment. On the basis of these transformations many kinematics of bodies can be built,

describing different physical properties, such as time dilation. It follows that there are infinitely

many different kinematics, which are consistent with the results of Michelson-Morley’s

experiments.

2. Adopted assumptions

The following assumptions have been adopted in the presented analysis:

I. Coordinate and time transformation «inertial frame of reference – inertial frame of reference»

is linear.

II. There is at least one inertial frame of reference in which the velocity of light in a vacuum is the

same in each direction. This system is called a universal frame of reference. This one-way

speed of light constant is indicated by the symbol c = constants.

III. The average velocity of light in the vacuum flowing way back and forth is constant for each

observer from the inertial frame of reference. This average velocity does not depend on the

observer’s velocity in relation to the universal frame of reference, nor on the direction of light

propagation. This average velocity is indicated by the symbol cp.

On the basis of assumption II and III it can be shown that the average velocity cp is equal to

one-way speed c. It is important to note that on the basis of III, the cp value is the same for each

observer, i.e. also for the one that does not move in relation to the universal frame of reference. As

for the motionless observer in relation to the universal frame of reference it has the value of c, thus

cp = c.

Let the light impulse move along a path of length L in one direction with velocity c+ ≥ 0 at

time t1, and in other direction along the same path L with velocity c– ≤ 0 at time t2. Then the average

velocity of light on the back and forth path is





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−+−+−

=

−+

=+

==

ccc

L

c

L

L

tt

Lccp 11

222

21

(1)

On this basis, an assumption III in the form of following equation is obtained

ccc

211=− −+ (2)

3. Transformations for one spatial dimension

Indications shown in Figure 1 are adopted. Coordinates in the universal frame of reference U

will be indicated with symbols x, t. Coordinates in the inertial frame of reference U' will be

indicated with symbols x', t'. The inertial frame of reference U' moves in relation to the universal

frame of reference U with velocity v along parallel x and x' axes. All velocities with direction such

as x axis direction (or x' in U' frame) have positive values, while those with opposite direction have

negative values. But the symbol c will always have a positive value, regardless of the direction in

which the light moves, i.e. always c = + 299 792 458 m/s.

Fig. 1. Inertial frame of reference U' move in relation to the universal frame of reference U with velocity v.

When the beginnings of frames overlapped, then the clocks at that beginning were reset to

zero. The clocks in universal frame of reference U were synchronized with the clock at the

beginning of this system with the light of Einstein method. At this stage, it is not determined how

the clocks in U' frame are synchronized.

Transformation from a universal frame of reference U to an inertial frame of reference U'

has, on the basis of assumption I, the following form

+=′

+=′

tfxet

tbxax

1

(3)

Transformation parameters are continuous velocity v functions with the following properties

]1[

]m/s[

[m/s]

]1[

)0)(0(

0)(

)0)(0(

0)(

1)0(

0)0(

0)0(

1)0(

1

>⇒<∧

>∧

<⇒>∧

>∧

=

=

=

=

vbv

vf

vbv

va

f

e

b

a

(4)

Parameters a(0) = 1, b(0) = 0, e1(0) = 0 and f (0) = 1 because for v = 0 the systems U and U'

are identical, i.e. they show the same coordinates of position and time.

The condition a(v) > 0must be met due to the same direction of x and x' axes (Figure 1), i.e.

if x increases, then x' also increases. The condition (v > 0 ⇒ b(v) < 0) must be met due to the same

direction of x and x' axes (Figure 1), i.e. for established x coordinate if time t elapse, then x'

coordinate decreases. Similarly, if the velocity v is negative, i.e. the U' frame moves in the opposite

t =t' = 0

U' v

0

x' 4

2

3

6 5

12 1

7

10

8

9

11

4

2

3

6 5

12 1

7

10

8 9

11

x U

1

0 1

t

t'





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direction, then coordinate x' increases, i.e. (v < 0 ⇒ b(v) > 0). Condition f (v) > 0 must be met as if t

increases, t' increases as well.

For our needs it will conveniently use the parameter e(v), where e1(v) = v·e(v). Introduction

of such a parameter is acceptable, because according to (4) there is an e1(0) = 0. The parameter e(v)

[s2/m

2] is a continuous velocity function v. Now the transformation (3) takes a form of

+=′

+=′

tfxvet

tbxax (5)

Differentials from transformation (5) have the form (v = constants) of

+=′

+=′

dtfdxvetd

dtbdxaxd (6)

Let’s consider the body, which rests in the inertial frame of reference U'. As it is motionless

in this system, and therefore for its coordinate of position there is

0=′xd (7)

Note that the velocity of body under consideration in relation to system U (i.e. dx/dt) is

a velocity v of system U' in relation to system U. Therefore

vdt

dx= (8)

From the differential coordinate (6) on the basis of (7) and (8) the following is obtained

bvabdt

dxa

dt

dtbdxa

dt

xdxd +=+=

+=

′=′=0 (9)

It follows that

vab −= (10)

On this basis the transformation (5) takes the form of

+=′

−=′

tfxvet

tvxax )( (11)


+=′

−=′

dtfdxvetd

dtvdxaxd )( (12)

Now consider a light impulse that moves along x and x' axes. When light moves in the direction of x' axis and velocity v, then the velocity of light in the inertial frame of reference U' has

the following value

)(vctd

xdx

+=′′

(13)

while in the universal frame of reference U has a value (assumption II) of

0≥= cdt

dx (14)





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When light moves in the direction opposite to x' axis and velocity v, then the velocity of light in the inertial frame of reference U' has the value of

)(vctd

xdx

−=′′

(15)

while in the universal frame of reference U has a value (assumption II) of

0≤−= cdt

dx (16)

When the differentials (12) are divided into sides, then on the basis of (13) and (14) the

following is obtained

fcve

vca

fdt

dxve

vdt

dxa

dtfdxve

dtvdxa

td

xdvcx +

−=

+

−=

+−

=′′

=+ )()()( (17)

When the differentials (12) are divided into sides, then on the basis of (15) and (16) the

following is obtained

fcve

vca

fdt

dxve

vdt

dxa

dtfdxve

dtvdxa

td

xdvcx +−

+−=

+

−=

+−

=′′

=− )()()( (18)

Formula (18) can be obtained from formula (17) by changing the sign before velocity c (this

means changing the direction of movement of a light pulse).

Formula (18) can also be obtained by simultaneously changing (17) the signs before

velocities v and cx+ (this means changing the direction of velocities v and x' axis direction at the

same time). In order to obtain formula (18), parameter e(v) must not change the sign. On this basis

the following properties are obtained for this parameter

0)()( ≥−⋅ veve (19)

From formulas (17) and (18) after taking into account (19), it results that one-way speed of light functions meet the following relation

),(),( cvccvc xx −−= +− (20)

),(),( cvccvc xx −= +− (21)

If dependencies (17) and (18) are added to equation (2) then it is obtained as follows

cvca

fcve

vca

fcve 2

)()(=

++−

+−+

(22)

In order for condition (2) to be met, parameter a(v) must be in the form of

)()()/(1

1 222

222

22 vefvef

cvvc

vefca +=+

−=

−+

⋅= γ (23)

On the basis of (23) the transformation (11) takes the form of





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+=′

−+=′

tfxvet

tvxvefx ))(( 22γ (24)

On this basis it is possible to determine the reverse transformation from the inertial frame of

reference U' to the universal frame of reference U in the form of

′+

+′+

−=

′+

+′+

=

tvef

xvef

vet

tvvef

xvef

fx

2222

2222

1

)(

1

)(

γ

γ (25)

Equations (24) and (25) are transformations sought for one spatial dimension. They contain

two parameters e(v) and f (v). These parameters must meet conditions (4) and (19). After the adoption of specific parameters, a specific transformation is obtained describing the specific

kinematics. This general form of transformation contains all possible linear transformations between the universal frame of reference U, in which light propagates with the constant velocity c,

and the inertial frame of reference U' moving relative to system U with velocity v, along x and x' axes, if in the inertial frame of reference U' of one-way speed of light has met condition (2) (i.e. assumption III).

4. Transformations for three spatial dimensions

In order to introduce two remaining spatial dimensions to the transformation, additional

assumption will be adopted:

IV. For each motionless observer in relation to the universal frame of reference, the space is isotropic, i.e. it has the same properties in each direction.

From assumption IV, results that parameters occurring in transformation (6) and (24)-(25)

meet the following properties

)()(

)()(

)()(

)()(

vfvf

veve

vbvb

vava

−=

−=

−−=

−=

(26)

Properties (26) result from (4), (19) and from the following reasoning. Parameters a(v) and f (v) must be even functions as if x' increases, then x' increases and if t increases then t' increases,

the same applies regardless of the direction of velocities v. The parameter b(v) must be odd function, because after a change of velocities v direction for a fixed coordinate x, if time t elapses,

the coordinate x' increases just as it did decreases for unchanged velocities v direction. The parameter v·e(v) must be odd because after a change of direction of velocities v for a fixed time t,

the change of time t' depends on x in the opposite way to unchanged direction of velocities v. Therefore, the parameter e(v) must be even function.

Consider the situation shown in Figure 2. In the inertial frame of reference U', the light impulse moves perpendicularly to x' axis. This light passes a distance of L', first in one direction

and then back, i.e. it returns to the starting point. Due to the assumption IV, the velocity of light in a direction perpendicular to x' axis is the

same in one direction and in the other and is c. This results from the fact that no direction





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perpendicular to velocities v (i.e. also x and x' axes) is distinguished (assumption IV) and the average velocity of light on the back and forth path is c (assumption III). For this reason, the same

light impulse for the motionless observer in relation to the universal frame of reference U will move along the arms of an isosceles triangle. For an observer from the system U, the dimensions

perpendicular to velocities v may be different than for an observer from the U' system, and therefore the height of triangle is determined by

LvL ′= )(ψ (27)

The parameter ψ (v) describes the transverse contraction of bodies moving in relation to the

universal frame of reference. This parameter should meet the following conditions

]1[0)(1)0( >∧= vψψ (28)

The parameter ψ (0) = 1 as for v = 0, the transverse dimensions are identical for observers

from U and U' systems. Condition ψ (v) > 0 must be fulfilled because the transverse dimensions do not reverse.

Fig. 2. The path of light seen from two frames of reference.

a) inertial frame of reference U', b) universal frame of reference U.

Due to assumption IV for the observer from the system U, the transverse dimensions are

contracted in the same way for each direction of velocities v. Therefore, parameter ψ (v) should meet the following condition

)()( vv −=ψψ (29)

Now the parameter ψ (v) will be set. For an observer from the system U', the following occurs

2

2 tcL

c

Lt

′∆=′⇔

′=′∆ (30)

On the basis of transformation (25) the differential (v = constants) is obtained.

x

x'

a)

U'

U

y

D

L=ψ (v)L' b)

v y'

L'

c, ½∆t'

c, ½∆t c, ½∆t

c, ½∆t'

½v∆t ½v∆t

D

t2 t1





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tdvef

xdvef

vedt ′

++′

+−

=2222

1

)(γ (31)

I.e. for the fixed coordinate x' in the system U' it is obtained as follows

tdvef

dtxd ′+

=⇒=′2

10 (32)

Formula (32) describes time dilation for motionless clock relative to U'. In Figure 2 such

a clock is at the beginning of system U'. If the time ∆t' on this clock elapses, appearing in formula

(30), then in system U will elapse the time ∆t = t2 – t1, where t1 is the time at which an impulse was sent, and t2 is the time when an impulse returned to x axis. Times t1 and t2 are measured in system U

by two different clocks. According to formula (32), the following occurs

tvef

t ′∆+

=∆2

1 (33)

From the geometry of figure it is obtained as follows

2222

4/ LtvD ′+∆= ψ (34)

and

c

Dt

2=∆ (35)

From equations (34) and (35) it is obtained as follows

c

Ltvt

22224/2 ′+∆

=∆ψ

(36)

)4/(4 222222 Ltvtc ′+∆=∆ ψ (37)

22222 )(4 tvcL ∆−=′ψ (38)

On the basis of (30) and (33) it is obtained as follows

2

22

2222

2

)(

1)(

44 t

vefvc

tc ′∆+

−=′∆

ψ (39)

222

222

)(

1

vefc

vc

+−

=ψ (40)

)()/(1

1

11)/(1

2

2

2

2

vefcv

vefcv

+−

=+

−=ψ (41)

That is, the transverse contraction parameter ψ (v) must have the following value

)(

12vef +

=γ

ψ (42)

The above analysis shows that formula (42) per parameter ψ (v) results from the assumption IV and time dilation (32).





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After taking into account (42) in transformation (24)-(25), the transformations for three spatial dimensions described by parameters e(v) and f (v) are obtained. From the universal frame of

reference U to inertial frame of reference U' has the following form

+=′

+=′

−+=′

+=′

zvefz

yvefy

tvxvefx

tfxvet

)(

)(

))((

2

2

22

γ

γ

γ (43)

From the inertial frame of reference U' to the universal frame of reference U has the

following form

′+

=

′+

=

′+

+′+

=

′+

+′+

−=

zvef

z

yvef

y

tvvef

xvef

fx

tvef

xvef

vet

)(

1

)(

1

1

)(

1

)(

2

2

2222

2222

γ

γ

γ

γ

(44)

The relation (42) can be written in a different way

21vef −=

ψγ (45)

After taking into account (45) in transformation (24)-(25) or (43)-(44), the transformations

for three spatial dimensions described by parameters e(v) and ψ (v)are obtained. From the universal

frame of reference U to inertial frame of reference U' the transformation has the following form

=′

=′

−=′

−+=′

zz

yy

tvxx

tvexvet

ψ

ψ

ψγ

ψγ

1

1

)(

1 2

(46)

From the inertial frame of reference U' to the universal frame of reference U, the

transformation has the form

′=

′=

′+′

−=

′+′−=

zz

yy

tvxvex

txvet

ψψ

ψγψγψ

ψγψ

22

2

(47)





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Equations (43)-(44) and (46)-(47) are transformations sought for all spatial dimensions. The

transformation set (43)-(44) is identical to the transformation set (46)-(47). These sets differ only in

the parameters used.

5. Selected transformations properties


=′

=′

−=′

−+=′

dzzd

dyyd

dtvdxxd

dtvedxvetd

ψ

ψ

ψγ

ψγ

1

1

)(

1 2

(48)


′=

′=

′+′

−=

′+′−=

zddz

yddy

tdvxdvedx

tdxdvedt

ψψ

ψγψγψ

ψγψ

22

2

(49)

5.1. Time dilation

The formula for time dilation for (46)-(47) transformation will be determined.

From the differential time (48) results that for the motionless observer in relation to

universal frame of reference U, the following formula for time dilation is given (also based on (45))

dtvfdtvetddx )(1

0 2 =

−=′⇒=

ψγ (50)

From the differential time (49) results that for the motionless observer in relation to inertial

frame of reference U', the following formula for time dilation is given

tddtxd ′=⇒=′ ψγ0 (51)

From formulas(50) and (51) results that observers from frames of reference U and U'

moving relative to each other will measure the same time dilation only if the parameter e(v) = 0. If

e(v) ≠ 0 then these two observers evaluate the relative passage of time differently on compared

clocks.

Time dilation (50) and (51) is written as an implication, as it is more precise than the record

commonly used in physics.





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5.2. Longitudinal length contraction (Lorentz-Fitz-Gerald)

The formulas for longitudinal length contraction (along x and x' axes) for transformation

(46)-(47) are determined.

From the differential coordinate (48), the following formula for longitudinal length

contraction (also based on (23) and (45)) results for the observer from the universal frame of

reference U

dxvadxxddt )(0 ==′⇒=ψγ

(52)

From the differential coordinate (49), the following formula for longitudinal length

contraction results for the observer from the universal frame of reference U

xdvedxtd ′

−=⇒=′ 220 ψ

γψ

(53)

From formulas (52) and (53) results that observers from frames of reference U and U'

moving relative to each other will measure the same longitudinal contraction only if the parameter

e(v) = 0. If e(v) ≠ 0 then these two observers evaluate differently the proportions of the longitudinal

dimensions measured by them.

Longitudinal length contraction (52) and (53) is written as implications, as this is more

precise than what is commonly used in physics.

5.3. Transformations of velocity

The formulas for transformations of velocity for transformations (46)-(47) will be

determined. Indications as in Figure 3 are adopted. The body moves in relation to U and U' systems.

For an observer from the system U it has velocity V, while for an observer from the system U' it has

velocity V'.

From equations (48) the following equations result

−+

=′′

−+

=′′

−+

−=

′′

dtvedxve

dz

td

zd

dtvedxve

dy

td

yd

dtvedxve

dtvdx

td

xd

2

2

2

1

1

1

1

1

)(

ψγ

ψ

ψγ

ψ

ψγ

ψγ

(54)

On this basis, the transformation of velocities from system U to system U' has the form of





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+−=′

+−=′

+−−

=′

1)(

1)(

1)(

)(2

vVve

VV

vVve

VV

vVve

vVV

x

zz

x

y

y

x

xx

ψγγ

ψγ

γ

ψγγ

(55)

Fig. 3. Movement seen from the universal frame of reference and inertial frame of reference.

From time equation (49) the following equations result

tdxdve

zd

dt

dz

tdxdve

yd

dt

dy

tdxdve

tdvxdve

dt

dx

′+′−

′=

′+′−

′=

′+′−

′+′

−

=

ψγψψ

ψγψψ

ψγψ

ψγψγψ

2

2

2

22

(56)

On this basis, the transformation of velocities from system U' to system U has the form of

ψγψψ

ψγψψ

ψγψ

ψγψγψ

+′−

′=

+′−

′=

+′−

+′

−

=

x

zz

x

y

y

x

x

x

Vve

VV

Vve

VV

Vve

vVve

V

2

2

2

22

(57)

Transformations of velocity (55) and (57) are equivalent. It is possible to show that after

introducing one to the other, the identity equations are obtained.

5.4. Velocity of light along axis x' seen in the inertial frame of reference

If the body shown in Figure 3 is an impulse of light, it moves with velocity c in system U.

Let’s consider only the case in which this impulse moves parallel to x and x' axes (i.e. it also moves

parallel to velocities v). Then

U

x

y U'

x'

V ′

U'

y' v

v

yV ′

xV ′

V yV

xVzV

z z'

zV ′

t t'





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0,0, === zyx VVcV (58)

On the basis of transformations (55), the velocity of this impulse of light seen in the inertial

frame of reference U' is obtained

0,0,1)(

)()(

2

=′=′=′=′+−

−=′=+

zzyyxx VcVcvcve

vcVvc

ψγγ

(59)

Formula (59) can also be obtained from formula (17) after applying dependencies (23) and

(45). Due to

vc

cvc

vcvc

cvc

vc

cvc

cvvc

+=−

−+=−

−=−

−=−

22

22

2

2

2 )())((

)()()/(1

1)(γ (60)

therefore, on the basis of (59) one-way speed of light with the same direction as x' axis direction

and velocities v direction has in inertial frame of reference U' the value of

0,0,)(2

2

=′=′++

=+zyx cc

vccve

cvc

γψ

(61)

One-way speed of light with the opposite direction to x' axis direction and velocities v

direction has a value in the inertial frame of reference U'

0,0,)(2

2

=′=′+−

=−zyx cc

vccve

cvc

γψ

(62)

Formula (62) is based on Formula (61) by changing the sign before velocity c (this means

changing the direction of movement of a light impulse). It can also be obtained by changing in

formula (61) the sign in front of velocity v (dependence (20) should then be taken into account,

which means a change in x' axis direction). Then before the parameters ψ (v), γ (v) and e(v), it is not

necessary to change the sign as these are even functions. Precisely because of such situations, it is

more convenient to use the even function e(v) used in transformation (5) than the odd function e1(v)

used in transformation (4).

5.5. Conclusions on one-way speed of light and the parameter e(v)

On the basis of (61) it is obtained

+

+ +−⋅=

x

x

ccv

vccce

2

2)(

ψγ

(63)

On the basis of (62) it is obtained

−

− −+⋅=

x

x

ccv

vccce

2

2)(

ψγ

(64)

On this basis, it is obtained

+

+

−

− +−⋅=

−+⋅

x

x

x

x

ccv

vccc

ccv

vccc2

2

2

2)()(

ψγ

ψγ

(65)





14 www.ste.com.pl

vcc

cvc

c

c

xx

−−=−+ +−

22

(66)

+

+

+−

−=−=

x

x

xx c

cccc

c

c

c

c 22

222

(67)

Ultimately, the relation between one-way speed of light in a vacuum parallel to x' axis is

obtained

+

+−

−=

x

xx

cc

ccc

2 (68)

From formulas (61) and (62) there is a valid conclusion on the parameter e(v). Let us check

for what values of the parameter e(v) it occurs, according to what was assumed in relation (2), i.e.

00 ≤∧≥ −+xx cc (69)

From (61) and (62) results that there must occur

≤+−⇒≤

≥++⇒≥

−

+

0)()(

)(0)(

0)()(

)(0)(

2

2

vccvvev

vvc

vccvvev

vvc

x

x

γψγψ

(70)

It follows that for inequalities (69) to occur, the parameter e(v) must meet the following

conditions

=+−

≤⇒≤

=−+−

≥⇒≥

−

+

max

min

])([)(

1)(0)(

])([)(

1)(0)(

vvevc

vc

cvvvevc

vvevc

vc

cvvvevc

x

x

ψ

ψ (71)

Fig. 4. Range of function values e(v)⋅v for which there is no apparent backward travel of light in time.

If conditions (71) are not met for some inertial system then the value of one-way speed of

light measured in this system is inconsistent with the time arrow. This means that the light may

-3 -2 -1 0 1 2 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

v [108 m/s]

e(v)⋅v [10–8

s/m]

[e(v)⋅v]max

[e(v)⋅v]min

STW: e(v)⋅v = –vγ (v)/c2

STE: e(v)⋅v = 0

cx–(v) = ±∞

cx+(v) = ±∞





15 www.ste.com.pl

seem to move back in time (its tide time is negative). This causes the velocity of light cx+(v) that is

consistent with x-axis direction to be negative, or the velocity of light cx–(v) that is not consistent

with x-axis direction to be positive. Then one of the inequalities will not be met (69). However,

equation (2) is still true, as it is more general than the one adopted at the beginning, for the purpose

of determining attention of inequalities (69).

Diagram 4 illustrates the inequalities (71). Functions [e(v)⋅v]min and [e(v)⋅v]max determine the

strip in which the function values e(v)⋅v must be located in order to prevent any inertial system from

experiencing the effect of apparent backward travel of light in time. This diagram also shows the

function values e(v)⋅v for STR (equation (75)) and STE (equation (84)).

Fig. 5. One-way speed of light in vacuum as a function of velocity v, for e(v)⋅v = –3vγ (v)/c2.

Figure 5 shows an example of how the values of one-way speed of light cx+(v) and cx

–(v)

depend on the speed of inertial system in which the observer is located when the parameter

e(v)⋅v = –3vγ (v)/c2.

It is advisable to consider a light impulse moving in the direction of x-axis. The impulse

moves from point x1 to point x2 > x1. Two clocks are required to measure one-way speed of light

cx+(v). The clock Z1 at point x1 measures the moment t1 when a light impulse is emitted. The clock Z2

at point x2 measures the moment t2 when a light impulse reaches it. In case shown in Figure 5, the

clocks in inertial systems are so desynchronized with parameter e(v) that in the inertial system

moving with velocity v = c/2 there is an equation t1 = t2. In this situation, the one-way speed of light

measurement gives an infinite value as

±∞=±

=−−

=+

0)2/(

12

12 L

tt

xxccx (72)

In fact, light in this inertial system has finite velocity because it has finite velocity in the

ether. The infinite velocity in (72) is caused by desynchronization of clocks. The clock Z2 is exactly

as late in comparison to the clock Z1 as the time it takes for the light to reach from point x1 to point

x2. Thus, equations (61) and (62) do not represent the actual velocity of light in a vacuum, but the

measurement result of this velocity using clocks that are desynchronized when e(v) ≠ 0.

In the inertial systems (Figure 5) moving at velocities v > c/2 the clocks are already so

desynchronized that the measurement of one-way speed of light cx+(v) gives negative values. This is

due to the fact that clock Z2 is late in comparison with clock Z1 for more time than the light needs to

reach from point x1 to point x2. In these inertial systems t2 – t1 < 0. Therefore, the light moving from

point x1 to point x2 apparently reverses in time.

-3 -2 -1 0 1 2 3-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

e(v)⋅v [10–8

s/m]

v [108 m/s]

[e(v)⋅v]max

cx–(v) = ±∞

[e(v)⋅v]min

cx+(v) = ±∞

e(v)⋅v = –3vγ (v)/c2

-3 -2 -1 0 1 2 3

-10

-8

-6

-4

-2

0

2

4

6

8

10

cx+(v), cx

–(v) [10

8 m/s]

v [108 m/s]

cx–(v)

cx–(v)

cx+(v)

cx+(v)





16 www.ste.com.pl

The fact that the moving back light in time is apparent, not real, will also be explained in

subsection 7.7. The significance of parameter e(v) is also discussed in Chapters 7.7, 8 and Błąd! Nie

można odnaleźć źródła odwołania..

6. General form of transformations expressed from one-way speed of light

In transformations (46)-(47), it is possible to replace the parameter e(v) with one-way speed

of light cx+(v) thanks to relation (63). Then the transformation from universal frame of reference U

to inertial frame of reference U' is obtained in the form of

=′

=′

−=′

+−⋅−+

+−⋅=′ +

+

+

+

zz

yy

tvxx

tvcc

vcccx

cc

vccct

x

x

x

x

ψ

ψ

ψγ

ψγ

ψγψγ

1

1

)(

)(1)(2

2

2

2

(73)

However, from the inertial frame of reference U' to the universal frame of reference U, the

transformation takes the form of

′=

′=

′+′

+−−=

′+′+−

−=

+

+

+

+

zz

yy

tvxvcc

vcccx

txcc

vccct

x

x

x

x

ψψ

ψγψγγψ

ψγψγ

2

2

2

2

)(

)(

(74)

In the same way, transformations (46)-(47) can be written on the basis of (64) using one-

way speed of light cx–(v).

Thanks to transformations (73)-(74), it is possible to define any transformation that meets

the I-IV assumption on the basis of two parameters, i.e. transverse contraction ψ (v) and one-way

speed of light in vacuum cx+(v).


the I-IV assumption on the basis of two parameters, i.e. transverse contraction ψ (v) and the

parameter of clock synchronization in inertial frames of reference e(v).


the I-IV assumption on the basis of two parameters, i.e. time dilation f (v) (resulting from(50)) and

the parameter of clock synchronization in inertial frames of reference e(v).

The importance of parameter e(v) is explained further in the article.





17 www.ste.com.pl

7. Special cases of transformation

7.1. Lorentz transformation – Special Theory of Relativity transformation

If we assume that

−−=−=

=

2

2

222 m

s1

)/(1

11)()(

]1[1)(

ccvcvve

v

γ

ψ (75)

then transformations (46)-(47) take the form of Lorentz transformation on which Special Theory of

Relativity is based. From the universal frame of reference U to the inertial frame of reference U',

the transformation takes the form of

=′−=′

=′

+−=′

zztvxx

yytxc

vt

)(

2

γ

γ (76)

From the inertial frame of reference U' to universal frame of reference U, the transformation

has the following form

′=′+′=

′=

′+′=

zztvxx

yytxc

vt

)(

2

γ

γ (77)

There are only two transformations (46)-(47), in which the corresponding coefficients in

transformation and in reverse transformation have the same value (with the accuracy to sign

resulting from the direction of velocities v). These are the Lorentz transformation and further shown

the Galilean transformation. For this reason, in Lorentz’s transformation the systems U and U'

become indistinguishable.

7.2. Lorentz transformations with transverse contraction

If we assume that

−−=−=

2

2

222m

s1

)/(1)(

11

)(

)()(

ccvvcv

vve

ψψγ

(78)

then transformations (46)-(47) take the form of transformation, which can be called Lorentz

transformations with transverse contraction. From the universal frame of reference U to the inertial

frame of reference U', the transformations take the form of

=′−=′

=′

+−=′

zztvxx

yytxc

vt

ψψγ

ψψγ

1)(

12

(79)

From the inertial frame of reference U' to universal frame of reference U, the

transformations have the following form





18 www.ste.com.pl

′=′+′=

′=

′+′=

zztvxx

yytxc

vt

ψψγ

ψψγ

)(

2 (80)

It should be explained where the idea of such a generalization of Lorentz’s transformation

came from in this article. In relation to formula (61) the one-way speed of light raises a natural

question, for which parameters e(v) and ψ (v) velocity of light in the inertial frame of reference will

have a value c in each direction. On the basis of (61), for light moving along the axis x', the

equation must be met

vccve

cc

++=

2

2

γψ

(81)

223 ccvccve =++γψ

(82)

I.e. for one-way speed of light to have the exact value c, the following must occur

2

1

ce

ψγ

−= (83)

It is easy to check on the basis of velocity transformations (55) that for all considered

transformations (46)-(47) the light moving parallel to y' axis will also have value c of one-way

speed in U' system. It also results directly from assumption IV, which is shown in Figure 2. Also

using numerical methods, it was verified for various cases of the function ψ (v) that one-way speed

of light in kinematics described by transformations (79)-(80) always has the value c.

It follows that there are infinitely many kinematics, in which one-way speed of light in

a vacuum, in every inertial frame of reference, is constant and is c. They are based on

transformations (79)-(80). The Special Theory of Relativity is only one of the infinite number of

such kinematics (transformation (76)-(77)).

In kinematics (79)-(80), in which we assume ψ (v) ≠ 1, the inertial frames of reference are

distinguishable, and there is a universal frame of reference, which was indicated by the symbol U.

The system U is distinguished by the way in which, according to transformation (46)-(47), the

lateral dimensions of bodies moving in relation to this system change. Therefore, such theories do

not meet the principle of equivalence of all inertial frames of reference.

In modern physics it is believed that one-way speed of light in a vacuum is absolutely

constant, i.e. it has the same value in every direction of propagation and for every observer. On this

basis STR Einstein was derived. It has been shown above that there are infinitely many kinematics

that meet this condition. The STR is distinguished as it additionally assumes the principle of

equivalence of all inertial frames of reference, i.e. that there is no such a physical phenomenon,

which distinguishes some inertial frame of reference. This comes down to the fact that the

corresponding coefficients in transformation and in reverse transformation must have the same

value (with the accuracy to sign resulting from the direction of velocities v). Among

transformations (79)-(80), only the Lorentz transformation (76)-(77) meets such an additional

assumption. However, there are no experimental grounds for assuming the principle of equivalence

of all inertial frames of reference. This principle has been introduced into physics in an arbitrary

manner.

Experimental evidence of the existence of a universal frame of reference is known. It is the

measurement of anisotropy of cosmic microwave background discussed in the Nobel dissertation

[7]. It turns out that electromagnetic cosmic microwave in a range of 300 GHz reaches from all





19 www.ste.com.pl

cosmic sides. Cosmic microwave in our frame of reference has dipole anisotropy. Cosmic

microwave reaching from the side of Lion constellation has a little more energy, while the one

reaching from the side of Aquarius constellation has a little less energy (Figure 6). If Doppler effect

is taken into account, it is possible to determine the frame of reference, in which cosmic microwave

background is homogeneous. Such a frame of reference is unique in relation to all others. The

existence of such a universal frame of reference suggests that even if one-way speed of light in

a vacuum was constant, the correct kinematics model is not the Special Theory of Relativity based

on Lorentz transformation (76)-(77), but a model based on some other transformation of (79)-(80)

form.

Fig. 6. Dipole anisotropy of cosmic microwave background

shown in Hammer-Aitoff projection (own elaboration based on [7]).

In the article [10] on the basis of Special Theory of Ether without transverse contraction, the

velocity of Solar System in relation to the system in which cosmic microwave background is

homogeneous is determined. A velocity of 369,3 km/s was obtained there (Figure 6), but the value

of this velocities will be different within other kinematics.

7.3. Transformations of Special Theory of Ether with transverse contraction

If we assume that

0)( =ve (84)

then transformations (46)-(47) take the form of transformations on which the Special Theory of

Ether is based with transverse contraction derived in the article [10]. From the universal frame of

reference U to the inertial frame of reference U', the transformations take the form of

=′−=′

=′=′

zztvxx

yytt

ψψγ

ψψγ1

)(

11

(85)



′=′+′=

′=′=

zztvxx

yytt

ψψγγψ

ψψγ (86)

galactic plane

K 0,0102,726 ±=vT mK 0,0173,358 ±=∆+ vT

mK 0,0173,358 ±−=∆− vT cv ⋅≈±= 0,001232km/s3,3369,3

center

of the galaxy





20 www.ste.com.pl

This is a whole class of transformations in which the simultaneity of events is absolute,

which means that the indications of clock do not depend on the coordinates of position. Kinematics

based on these transformations differ in physical properties, e.g. transverse contraction and time

dilation.

7.4. Transformation of Special Theory of Ether without transverse contraction

If we assume that

0)(

1)(

=

=

ve

vψ (87)

then transformations (46)-(47) take the form of transformation on which the Special Theory of

Ether is based without transverse contraction derived in the article [9]. From the universal frame of

reference U to the inertial frame of reference U', the transformation takes the form of

=′−=′

=′=′

zztvxx

yytt

)(

1

γγ (88)



′=′+′=

′=′=

zztvxx

yytt

γγ

γ1 (89)

In this case of the Special Theory of Ether, transverse contraction does not occur (czyli

ψ (v) = 1). The Special Theory of Ether derived based on transformation (88)-(89) is closely linked

to the Special Theory of Relativity by Einstein. This was proven in the work [8].

The transformation (89) was already derived in articles [2], [4] by another method. In those

articles the authors obtained such transformation from the Lorentz transformation thanks to the

synchronization of clocks in inertial frames of reference by the external method. The transformation

obtained in the works [2], [4] is the Lorentz transformation differently written down after a change

in the manner of time measurement in the inertial frame of reference, this is why the properties of

the Special Theory of Relativity were attributed to this transformation. In the article [9] the

transformation (88)-(89) has a different physical meaning than the Lorentz transformation, because

according to the theory outlined in that article, it is possible to determine the speed with respect to

a universal frame of reference by measurement. So the universal frame of reference is real, and this

is not a freely chosen inertial frame of reference.

7.5. Transformation of Special Theory of Ether with absolute time

If we assume that

0)(

1)/(1)(/1)( 2

=

≤−==

ve

cvvv γψ (90)





21 www.ste.com.pl

then transformations (46)-(47) take the form of transformation in which the absolute time occurs.

From the universal frame of reference U to the inertial frame of reference U', the transformation

takes the form of

=′−=′

=′=′

zztvxx

yytt

γγ

γ

)(2

(91)



′=′+′=

′=′=

zztvxx

yytt

γγ

γ11

1

2

(92)

In kinematics based on this transformation, time elapses the same way in all inertial frames

of reference, just like in Galilean transformations. It is very interesting that the theory with the

absolute time which meets the conditions of Michelson-Morley and Kennedy-Thorndike

experiments is possible.

7.6. Transformation of Special Theory of Ether without longitudinal contraction

If we assume that

0)(

1)/(1/1)()( 2

=

≥−==

ve

cvvv γψ (93)

then transformations (46)-(47) take the form of transformation on which the Special Theory of

Ether is based without longitudinal contraction. From the universal frame of reference U to the

inertial frame of reference U', the transformation takes the form of

=′−=′

=′=′

zztvxx

yytt

γ

γγ1

112

(94)



′=′+′=

′=′=

zztvxx

yytt

γγ

γγ2

2

(95)

In kinematics based on this transformation the longitudinal dimensions (parallel to x and x'

axes) are the same for observers from every inertial frame of reference. This is due to differentials

from transformation (94)-(95)

tdvxddx

dtvdxxd

′+′=

−=′2γ

(96)

I.e.





22 www.ste.com.pl

xddxtd

dxxddt

′=⇒=′

=′⇒=

0

0 (97)

7.7. Extended Galilean transformations and conclusions on the parameter e(v)

If we assume that

)1)(0/(1)(

1)/(1/10/ 2

≅⇒→∨=

≅−=⇒→

vcvv

cvcv

ψψ

γ (98)

then transformations (46)-(47) take the forms that can be called the extended Galilean

transformations. From the universal frame of reference U to the inertial frame of reference U', the

transformations take the form of

=′−=′

=′−+=′

zztvxx

yytvexvet )1( 2

(99)



′=′+′−=

′=′+′−=

zztvxvex

yytxvet

)1(2

(100)

The transformations (99)-(100) applies when velocity v is very small in relation to c, or

simply when c = ∞. Then on the basis of (61) and (62) the one-way speed of light flowing along x-

axis is

∞=

== −+

c

vvevcvc xx

)(

1)()(

(101)

On the basis of dependency (101), important conclusions can be drawn. One-way speed of

light in a vacuum has an infinite value (c = ∞) in relation to the universal frame of reference. Then,

of course, this velocity has an infinite value in every inertial system. However, according to (101)

the one-way speed of light in the inertial system is 1/(e v). This seems to be contradictory.

Fig. 7. Measurement of the velocity of light in an inertial system when the clocks are desynchronized in the system

and when the light in the universal frame of reference (ether) has infinite velocity.

However, there is no contradiction in this. It should be noted that in order to measure one-

way speed it is necessary to use at least two motionless clocks in the system in which the

measurement is carried out. Different items of clocks are used in the universal frame of reference

than in the inertial system. If in the inertial system the clocks are desynchronized, the measurement

c = ∞

4

2

3

6 5

12 1

7

10

8

9

11

U'

c = –∞

4

2

3

6 5

12 1

7

10

8 9

11

4

2

3

6 5

12 1

7

10

8 9

11

U - ether 4

2

3

6 5

12 1

7

10

8

9

11

t'0 = 0 t'1 ≠ 0

t0 = 0 t1 = 0

a)

b)

x' = 0 x' = D'

x = 0 x = D

v





23 www.ste.com.pl

of light speed moving with infinite velocity can give a seeming finite value. This has been

illustrated in Figure 7.

In part b) of the figure, the universal frame of reference is shown. The light is sent from

point x = 0 at the moment t0 and immediately reaches point x = D at the moment t1. As it is assumed

in the universal frame of reference, the velocity of light is infinite, t0 = t1 = 0.

In part a) of the figure, the inertial system U' is shown. At the moment under consideration,

the two clocks of this system are located directly next to the two clocks of system U. Due to length

contraction, the distance of clocks measured in system U' is D' and may have a different value than

D measured in system U. The clock at point x' = 0 indicates the time t'0 = 0, but the clock at point

x' = D' indicates another time t'1 ≠ 0. According to the observer from system U', the light was sent at

the moment t'0 = 0, while to point D' it reached at the moment t'1 ≠ 0. Therefore, in system U' the

one-way speed of light is

−∞>

∞<

′−′′

=+

01

)(tt

Dvcx (102)

The velocity (102) is the result of a measurement carried out by desynchronized clocks in

system U'. It is not a true velocity of light, which in this case has an infinite value. What’s more, the

clocks in the inertial system U' can be so desynchronized that the light will move back in time. It

will be so when t'1 < t'0. Such cases were discussed in Chapter 5.5.

A similar situation is obtained when the light moves in the opposite direction, i.e. from point

x = D to point x = 0.

The situation shown in Figure 7 occurs for transformations where the parameter e(v) ≠ 0

(e.g. in the Special Theory of Relativity). Then the clocks passing next to each other already at the

moment of their synchronization indicate other values. The above example shows that then the one-

way speed of light (61) and (62) measured in inertial systems do not reflect the real velocity of light

in these systems.

Therefore, the parameter e(v) ≠ 0 causes desynchronization of clocks. Readings from such

clocks should not be taken literally, and theories based on such a parameter, such as STR, should be

interpreted differently than it is done in modern physics. This subject will be further elaborated in

Chapter 8.

In case of the extended Galilean transformations in the inertial frame of reference U' (but not

in system U ), the clock indications have been rearranged (the clocks have been desynchronized) to

the natural setting occurring in Galilean transformation. For this reason, in time transformations

there is a factor depending on the position x or x'.

7.8. Galilean transformation

If we assume that

)0)(0/(0)(

)1)(0/(1)(

1)/(1/10/ 2

≅⇒→∨=

≅⇒→∨=

≅−=⇒→

vecvve

vcvv

cvcv

ψψ

γ

(103)

then transformations (46)-(47) take the form of Galilean transformation on which classical

kinematics is based. From the universal frame of reference U to the inertial frame of reference U',

the transformation takes the form of





24 www.ste.com.pl

=′−=′

=′=′

zztvxx

yytt (104)



′=′+′=

′=′=

zztvxx

yytt (105)

In Galilean transformation, the corresponding coefficients in transformation and in reverse

transformation have the same value (with accuracy to the sign resulting from the direction of

velocities v). For this reason, in the Galilean transformation, as in the Lorentz transformation, the U

and U' systems become indistinguishable.

The Galilean transformation can be treated as an approximation of all linear transformations

derived in this article for small velocities v, i.e. when v << c. Therefore, classical kinematics is

consistent with experiments on small velocities v regardless of which of the infinitely many possible

kinematics is the best model of real processes.

The transformation (104)-(105) applies when velocity v is very small in relation to c, or

simply when c = ∞. Then on the basis of (61) and (62) the one-way speed of light flowing along x-

axis is

∞==−= −+cvcvc xx )()( (106)

8. Physical significance of parameters occurring in transformations and

discussion of relativity basics

8.1. Parameters f (v), a(v) and ψ (v)

From formulas (50) and (51) for time dilation, it results that the parameter f (v), which occurs

in transformation (43)-(44), describes time dilation. For the motionless observer in relation to

universal frame of reference, the time in inertial frame of reference elapses f (v) times faster (1/f (v)

times slower) than in its universal frame of reference.

From formulas (52) and (53) for longitudinal contraction it results that parameter a(v),

which occurs in transformations (5) and (11), describes longitudinal contraction (i.e. parallel to

velocities v) of bodies in motion in relation to bodies resting in relation to the universal frame of

reference U. For the motionless observer in relation to universal frame of reference, the moving

body is a(v) times shorter (1/a(v) times longer) than the same motionless body in relation to the

universal frame of reference.

The parameter ψ (v), which occurs in transformation (46)-(47), describes transverse

contraction (perpendicular to velocities v) of bodies moving in relation to bodies resting in relation

to the universal frame of reference U (Figure 2). That is, the moving body is ψ (v) times wider

(1/ψ (v) times narrower) than the same motionless body in relation to the universal frame of

reference.





25 www.ste.com.pl

8.2. Parameter e(v) = 0

On the basis of transformations (43)-(44) and (46)-(47) it can be concluded that parameter

e(v) can be treated as a way of clock synchronization in the inertial frames of reference. Consider

the case when e(v) = 0. Then the time transformation (85), from which the following is obtained

)00(1

=′⇒=⇒=′ ttttψγ

(107)

This means that for e(v) = 0, the clock synchronization in system U' consists in the fact that

if the clock of system U indicates time t = 0, then according to (107), the clock of system U' next to

it is also reset to zero, that is t' = 0. This way of synchronization is shown in Figure 8.

Fig. 8. External clock synchronization (e(v) = 0).

The clocks in universal frame of reference U were synchronized with light, which in this system has one-way speed c.

In the time under consideration, when all clocks of the system U indicate time t = 0, the beginnings of systems U and U'

coincide. On each clock passing by clock t = 0 the time is also set t' = 0.

In this particular case, when e(v) = 0, one-way speed of light (61) and (62) take the

following values

vc

cvcve x +

=⇒= +2

)(0)( (108)

vc

cvcve x −

−=⇒= −2

)(0)( (109)

8.3. Parameter e(v) ≠ 0

Let’s consider cases for any parameter e(v). Transformation of time (47) has the form of

txvet ′+′−= ψγψ 2 (110)

When the clocks in system U indicate time t = 0, then according to Figure 1 the beginnings

of systems coincide. From equation (110) it results that on the clock from system U' located next to

zeroed clock from system U, the value t' is set, which is expressed by the formula

xve

tt ′=′⇒=γ

ψ0 (111)

Synchronization of clocks in the system U' is shown in Figure 9. From the perspective of

system U, clocks in the system U' are desynchronized, as their indications depend on the position

and not only on the time lapse. If the observer from system U' measures one-way speed of light,

U'

x' 0

0

v

t = 0

t' = 0 t' = 0 t' = 0

t = 0 t = 0

U

x c

t =0 ⇔ t' =0 t =0 ⇔ t' =0 t =0 ⇔ t' =0

c

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then on some clock reads the initial time, while on another clock measures the final time. These

clocks can always be set in such a way (desynchronize them) that one-way speed of light will have

the predetermined value of cx+(v) and cx

–(v). The same effect can also be achieved in classical

mechanics. If to move forward the indications of clock from which the final time is read, the

seemingly velocity of body will be smaller, while if to move back the indications of this clock, the

seemingly velocity of body will be larger.

Fig. 9. External clock synchronization (e(v) ≠ 0).

The clocks in universal frame of reference U were synchronized with light, which in this system has one-way speed c.

In the time under consideration, when all clocks of the system U indicate time t = 0, the beginnings of systems U and U'

coincide. On each clock passing by clock t = 0 the set time t' = ψevx'/γ.

For considered transformations, if the clocks in system U' are set according to formula (111)

then one-way speed of light will have the value expressed in formulas (17), (18) and (61), (62).

However, this does not mean that it is a velocity resulting from the speed at which the real

processes on which the clocks are based take place. This can only be the result of setting the clocks

in inertial frames of reference as shown in Figure 9. As kinematics model should be expected to

describe real processes, clocks in inertial frames of reference cannot be set freely, only in a way that

corresponds to the described processes.

If e(v) ≠ 0, then formulas (50) and (51) are different, as well as formulas (52) and (53) are

different. Therefore, observers from U and U' systems draw different conclusions on time dilation

and longitudinal contraction on the basis of their measurements (they evaluate the relative passage

of time in their systems differently and they evaluate the proportions of horizontal lines in their

systems differently). Such a situation can be interpreted in such a way that their measuring devices

have not been synchronized and for this reason they measure something else. Only if e(v) = 0, then

their measurements of time dilation and longitudinal contraction give the same result, i.e. only then

the clocks with their frames of reference were correctly synchronized.

Consider the light impulse sent to the right from the beginning of system U' in time

synchronization of clocks (Figure 9). In time t'1 = 0, the impulse was in position x'1 = 0, while in time

t'2 it was in position x'2. On the basis of (61), it is possible to write that

vc

c

xx

vex

c

vccve

vc

x

vc

xxttt

xx

+

′+′=′

++=

′=

′−′=′=′−′ ++ 2

2222

2

212212

)()( γψγ

ψ

(112)

Formula (112) may have different interpretations. In the Special Theory of Relativity, the

interpretation is that clocks in the inertial frame of reference U' are correctly synchronized.

Therefore, the light actually needed t'2 time to travel the distance of x'2. Then actually for the

observer from system U' the light has one-way speed expressed by the formula (61). For STR on the

basis of (75) this velocity has the value of

U'

x' 0

0

v

t = 0

t' =ψevx'1 /γ t' = 0

t = 0 t = 0

U

x c c

cx–(v) cx

+(v) cx–(v) cx

+(v)

x'1

t' =ψevx'2 /γ

x'2

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c

vccvc

c

vccve

cvcx =

++−=

++=+

2

2

2

2

2

11)(

γγγ

ψ (113)

However, it is important to remember that the values t'2 and t'1 = 0 are read from two different

clocks. If these clocks are not synchronized correctly, then the velocity (113) is apparent. Then the

velocity (61) does not reflect the course of physical processes, but is caused by the way clocks are

set in the inertial frame of reference U'. For such an interpretation, after taking into account (108),

the formula (112) will be written in the following form

)0(

222

22 =

′=

+

′=′−′ +

ec

x

vc

c

xx

vet

xγψ

(114)

That is, when a light impulse was emitted, then the value (111) was set on the clock at point

x'2, but in reality the value of 0 resulting from the correct synchronization, i.e. from formula (107),

should be set. Therefore, when the impulse reaches point x'2, then the correct indication of a clock in

that point is not t'2, but only

22 xve

t ′−′γ

ψ (115)

With this interpretation, the left side of equation (114) is the real time that an impulse

needed to reach point x'2. If the clock at point x'2 is correctly synchronized according to the formula

(107), then the one-way speed of light will be (108) or (109), and not (61) or (62).

It follows from the above that if the parameter e(v) ≠ 0, then different interpretations of

transformation (43)-(44) and (46)-(47) are possible. In the Special Theory of Relativity the

interpretation is that readings from the clocks should be treated literally in this situation. This leads

to the fact that different observers measuring the same physical phenomena obtain different results

(except for one-way speed of light in a vacuum). In STR it was considered to be a property of

space-time and not a result of synchronization of clocks between inertial frames of reference.

For the second interpretation of the parameter e(v), adopting that e(v) ≠ 0 causes the clocks

to be desynchronized in the inertial frame of reference, but all the time it is the same kinematics as

the one based on parameter e(v) = 0. After the clocks are desynchronized the values shown by these

clocks should not be treated literally. If the calculation takes into account the fact that clocks are

desynchronized, then each kinematics with the parameter e(v) ≠ 0 comes down to kinematics with

the parameter e(v) = 0. According to this interpretation, the parameter e(v) does not allow other

kinematics to be obtained. All kinematics possible for the I-IV assumptions adopted in this article

are included in transformations (85)-(86). Kinematics differ in only one parameter of transverse

contraction ψ (v). In monograph [8] it is shown, that with such an interpretation STR becomes STE

with a universal frame of reference. According to this interpretation, numerous conclusions of

modern physics drawn from the mathematics on which STR is based are incorrect. Therefore, STR

mathematics is correct, but the interpretation of this mathematics is not correct.

Introduction of parameter e(v) ≠ 0 for the Galilean transformation (104)-(105) leads to

transformation (99)-(100). This means that after desynchronization of clocks between different

inertial frames of reference, a transformation is obtained in which the values indicated by the clocks

of system U' depend on their position. However, it is still classical kinematics, only written in

a more complex way. After all, the way of setting the initial values on the clocks of system U' has

no influence on the course of physical processes in classical mechanics. However, it is formally

possible to write this kinematics with the parameter e(v) ≠ 0. If in classical kinematics written with

transformation (99)-(100), the values indicated by desynchronized clocks are treated literally, then





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there are similar conclusions to those drawn from the Lorentz transformation in the Special Theory

of Relativity. That is, for example, that the simultaneity of events, time dilation and longitudinal

contraction are relative. However, if the second interpretation is adopted, then all kinematics (99)-

(100) come down to the classical kinematics described by transformations (104)-(105).

In this article, the interpretation of parameter e(v) was adopted in such a way that it

describes the desynchronization of clocks in inertial frames of reference. In monograph [8], in

chapter “What is the Special Theory of Relativity (STR)” it was shown that the parameter e(v) can

be assigned another, third interpretation. The parameter e(v) can describe displacement in time and

space that the transformation implements. It is commonly believed that the transformation is related

to clocks that are directly adjacent to each other in a given time. That is, it recalculates coordinates

of the same event seen from different frames of reference. This is how Lorentz’s transformation in

the Special Theory of Relativity is understood. But the transformation can calculate the coordinate

of clock’s position to the coordinate of the same clock in another frame of reference, but one at

which the clock will be in the future or was in the past. With this interpretation, the transformation

does not calculate the coordinates of the same event, but the coordinates of different events. With

this interpretation of parameter e(v) the transformation is related to coordinates of the same event

only if e(v) = 0.

9. Importance of parameter e(v) for time dilation

In case of transformations, in which the parameter e(v) ≠ 0 is not possible to state directly

from the clock readings that in some inertial system time elapses slower or faster than in universal

frame of reference (or other inertial system).

Let us consider the situation shown in Figure 10. In the inertial system U' there is an

observer O', while in the universal system U there is an observer O.

Fig. 10. Times measured by two observers from different reference systems.

The observer O' has in his system Bi clocks. The clock B0 is located directly next to him.

The observer O has in his system Ai clocks. The clock A0 is located directly next to him. Observers

O and O' cannot directly compare A0 and B0 clocks because they are far apart and in constant

relative motion.

Each of these observers can at any time read the time from the two clocks that are directly

next to him. The observer O can read the time from his A0 clock and from Bi clock passing by. From

a particular Bi clock he can read the time only once, when that particular clock is next to him. Each

time he reads the time from Bi clock, it is a different clock.

x', t' v

U - ether

U'

O

O'

x, t

B0 B1 B2 B3

A3 A2 A1 A0

these clocks fly next

to the observer O

these clocks fly next

to the observer O'

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The same applies to the observer O'. The observer O' can read the time from his B0 clock

and from Ai clock passing by. He can read the time from a specific Ai clock only once when that

particular clock is next to him. Each time he reads the time from Ai clock, it is a different clock.

The observer can read the time from the clock that is currently next to him. He can also read

the position of such a clock according to the coordinates of U and U' systems. The coordinate and

time transformation derived in this paper are used to convert readings carried out according to one

reference system to readings carried out according to another reference system.

In the situation under consideration, the time dilation equation (50) applies to the observer O

from the universal system, while the time dilation equation (51) applies to the observer O' from the

inertial system. In other words, the observer O evaluates the relative time elapse on the clocks he

compares in the following way

210 ve

dt

tddx −=

′⇒=

ψγ (116)

In turn the observer O' evaluates the relative elapse of time on the compared clocks in the

following way

ψγ1

0 =′

⇒=′dt

tdxd (117)

This means that if e(v) ≠ 0, then the observers O and O' will evaluate the relative time elapse

differently (this has already been noted in subsection 5.1). It should also be noted that the observer

O reads the time from clocks other than the observer O'.

For the observer O from the universal system, the time elapsed on Bi clocks is not measured

by one clock, but by many clocks passing one after the other. If Bi clocks are desynchronized, then

the observer O reading from them does not measure the real time elapse in U' system. In equation

(116) there is the factor e(v)⋅v2, which describes how the desynchronization of Bi clocks affects the

time dilation, measured by the observer O.

The observer O' from the inertial system evaluates time dilation on the basis of Ai clocks.

These clocks are motionless in relation to the ether and have been desynchronized by means of

light, which in assumption has one-way speed in the ether with a constant value of c. The

dependence (117) shows that parameter e(v) does not affect time dilation measurement, which is

performed by the observer O'. Therefore, Ai clocks were not desynchronized with this parameter.

The observer O' reads the real time elapse in the universal frame of reference U.

If e(v) ≠ 0, then the time dilation measurements by the observer O and O' are different.

Therefore, it is not objectively possible to state that time elapses faster in one system and slower in

another. However, it can be stated that their clocks are not desynchronized.

The fact that parameter e(v) describes how to desynchronize clocks in the inertial systems

can also be deduced from the time dilation dependency (50). The observer O, shown in Figure 10, is

motionless relative to the universal frame of reference U and evaluates the time elapse in its U

system on the basis of one A0 clock. As all clocks measure time according to the time arrow, the

time elapse on A0 clock meets the condition dt > 0. Based on (50) the following is obtained

0)()(

1)(00

2<′⇒

>∧>∧= td

vvvvedtdx

ψγ (118)

In other words, the observer O reads the time on successive Bi clocks and states that a later

reading indicate an earlier, not later, moment of time. If the observer O treated such readings

literally, as is currently done in the Special Theory of Relativity, he would conclude that in the

inertial system U' time reverses. However, this is not true, because by definition each Bi clock





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measures time according to the time arrow. The reason for such a strange measurement is that the

observer O when reading the time on Bi clocks does not measure the actual time elapse in the

inertial system U'. Its measurement is influenced by how the clocks of U' system are

desynchronized by the parameter e(v). That is, if Bi+1 clock is very late in relation to Bi clock

(ti+1 << ti), then the time elapse that the observer O will measure on Bi and Bi+1 clocks will be

negative (ti+1 – ti < 0). The way the clocks are desynchronized is shown in Figure 9. In order for this

effect to occur, the parameter e(v) must has positive and enough large values. This has been shown

in Figure 11.

Fig. 11. Influence of parameter e(v) on time measurement in a mobile inertial system by an observer with universal

frame of reference (dt > 0).

This issue has been addressed in another way in Chapter 7.7, where, on the example of

Galilean extended transformations, it has been shown that evaluating the time elapse on the basis of

two different clocks, even if they are from the same inertial system, leads to misleading conclusions

if the parameter e(v) ≠ 0. In this case, if the observer reads t0 on one clock and t1 on another, it does

not mean that time t1 – t0 passed between readings. The difference t1 – t0 is affected by the actual

time elapse, but also by the extent to which the two clocks are desynchronized.

10. Conclusion

The article presents the original method of studying the transformations of time and

coordinate in terms of acceptable interpretations that can be attributed to these transformations.

All possible linear transformations that meet the results of Michelson-Morley’s and

Kennedy-Thorndikea’s experiments (without turnovers) have been derived in this article. On the

basis of these transformations it is possible to build numerous kinematics with different physical

properties. Thus, there are infinitely many kinematics consistent with experiments in which velocity

of light was measured.

For each kinematics an infinite number of dynamics can be derived. The method that allows

this is shown in the papers [8] and [12].

In this article it was also shown that there are infinitely many different kinematics, in which

one-way speed of light in a vacuum has in each direction and in each inertial frame of reference the

e(v)⋅v [10–8

s/m]

-2 -1 0 1 2-5

-4

-3

-2

-1

0

1

2

3

4

5

v [108 m/s]

-3 3

e(v)⋅v=1/(γ (v)ψ (v) v)

dt' < 0

dt' < 0

dt' > 0

dt' = 0





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value of c (transformations (79)-(80)). The Special Theory of Relativity is only one of those

infinitely many kinematics.

The phenomenon of dipole anisotropy of cosmic microwave background proves that there is

a universal frame of reference, in which this cosmic microwave background is homogeneous. This

shows that in reality the inertial frames of reference are experimentally distinguishable, i.e. they are

not equivalent. It follows that even if we assume that one-way speed of light is absolutely constant,

the Special Theory of Relativity is not the correct model of real processes. If one-way speed of light

is absolutely constant, then the correct model of real processes will be kinematics based on one of

the Lorentz transformation with transverse contraction (79)-(80).

Kinematics, in which one-way speed of light is always constant, are possible if one assumes

the interpretation of parameter e(v) as adopted in the Special Theory of Relativity, i.e. that they

describe specific properties of space-time and not the desynchronization of clocks in inertial frames

of reference.

However, the parameter e(v) can be interpreted differently, i.e. in such a way that it

describes the way of desynchronizing clocks in inertial frames of reference in relation to the

universal frame of reference. In this article the thesis was formulated that assuming the parameter

e(v) ≠ 0 leads to desynchronization of clocks between different inertial frames of reference.

However, the way of setting the initial values on clocks located in inertial frames of reference does

not affect the physical processes. If such an interpretation of this parameter is assumed, then any

kinematics with parameter e(v) ≠ 0 comes down to kinematics based on parameter e(v) = 0.

Therefore, the only parameter that can differ from kinematics meeting the I-IV assumptions is the

transverse contraction parameter ψ (v). Then it is not possible kinematics in which one-way speed

of light is constant in every inertial frame of reference. All kinematics are included in

transformations (85)-(86). The one-way speed of light formula for these kinematics has been

derived in this article [10].

The paper shows that interpretation of parameter e(v) as assumed in STR leads to

contradiction (Chapter 7.7). Therefore, the commonly accepted in physics interpretation of this

parameter is mistaken. The correct interpretation of this parameter is that if e(v) ≠ 0, then the clocks

in inertial systems are desynchronized. Then it is not possible to compare readings from different

clocks of the same inertial system literally.

Regardless of how the parameter e(v) is interpreted, parameter ψ (v) is not a variation of

scale. This is obvious, if it is noticed that in every inertial system the measurements are carried out

using technically identical devices. Firstly, in some inertial system the identical clocks for time

measurement and identical rulers for distance measurement are generated. Then some of these

devices are transferred to other inertial systems. Transformations derived in this article describe the

relations between measurements carried out by such identical devices placed in the universal frame

of reference and various inertial systems. Any change to parameter ψ (v) changes the physical

properties of kinematics. It is sufficient to note that this parameter determines the time dilation

described by the formula (51). That is, the way in which time is measured by clocks moving in

relation to the universal frame of reference depends on the value of parameter ψ (v).

All experiments conducted by man were observed with inertial frames of reference moving

with small velocities relative to universal frame of reference. Such experiments do not provide an

answer on how the laws of nature look like for observers found in the inertial frames of reference

moving with large velocities relative to universal frame of reference. Therefore, in physical

theories, the results obtained in frames of reference available to the observer are extrapolated to all

other inertial frames of reference. But as, they are acceptable as valid models of real processes,

kinematics based on transformations that do not meet the III assumption in all inertial frames of

reference, but only in inertial frames of reference available for experiments. The introduction of

such transformations is presented in the article [13].





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[2] Mansouri Reza, Sexl Roman U., A Test Theory of Special Relativity: I. Simultaneity and Clock

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pomocy teorii z eterem (in Polish), viXra 2017, www.vixra.org/abs/1704.0302.

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отсчета (in Russian), viXra 2018, www.vixra.org/abs/1806.0198.

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